Abstract interpretation can be applied to speed up constrained combinatorial search. The approximations used are supersets of the constraints. The authors study an instance of this method by selecting as the supersets the cylindrical closures of Ashby (1964). The result is the method of pruning by cylindrical closure (PCC). When the closures are of order 1 and the constraints are binary, the arc consistency method of Mackworth (1977) is obtained. In the nonbinary case one gets the lookahead inference of CHIP. PCC with closure of order higher than 1 yields new consistency methods